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				<span dir="auto">Z-transform</span>
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				<div id="mw-content-text" dir="ltr" class="mw-content-ltr" lang="en"><p>In <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a>, the <b>Z-transform</b> converts a <a href="http://en.wikipedia.org/wiki/Discrete_mathematics" title="Discrete mathematics">discrete</a> <a href="http://en.wikipedia.org/wiki/Time-domain" title="Time-domain" class="mw-redirect">time-domain</a> signal, which is a <a href="http://en.wikipedia.org/wiki/Sequence" title="Sequence">sequence</a> of <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real</a> or <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex numbers</a>, into a complex <a href="http://en.wikipedia.org/wiki/Frequency-domain" title="Frequency-domain" class="mw-redirect">frequency-domain</a> representation.</p>
<p>It can be considered as a discrete-time equivalent of the <a href="http://en.wikipedia.org/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>. This similarity is explored in the theory of <a href="http://en.wikipedia.org/wiki/Time_scale_calculus" title="Time scale calculus" class="mw-redirect">time scale calculus</a>.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
<span class="toctoggle">&nbsp;[<a href="#" class="internal" id="togglelink">hide</a>]&nbsp;</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#History"><span class="tocnumber">1</span> <span class="toctext">History</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Definition"><span class="tocnumber">2</span> <span class="toctext">Definition</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Bilateral_Z-transform"><span class="tocnumber">2.1</span> <span class="toctext">Bilateral Z-transform</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Unilateral_Z-transform"><span class="tocnumber">2.2</span> <span class="toctext">Unilateral Z-transform</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#Geophysical_definition"><span class="tocnumber">2.3</span> <span class="toctext">Geophysical definition</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-6"><a href="#Inverse_Z-transform"><span class="tocnumber">3</span> <span class="toctext">Inverse Z-transform</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="#Region_of_convergence"><span class="tocnumber">4</span> <span class="toctext">Region of convergence</span></a>
<ul>
<li class="toclevel-2 tocsection-8"><a href="#Example_1_.28no_ROC.29"><span class="tocnumber">4.1</span> <span class="toctext">Example 1 (no ROC)</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Example_2_.28causal_ROC.29"><span class="tocnumber">4.2</span> <span class="toctext">Example 2 (causal ROC)</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Example_3_.28anticausal_ROC.29"><span class="tocnumber">4.3</span> <span class="toctext">Example 3 (anticausal ROC)</span></a></li>
<li class="toclevel-2 tocsection-11"><a href="#Examples_conclusion"><span class="tocnumber">4.4</span> <span class="toctext">Examples conclusion</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-12"><a href="#Properties"><span class="tocnumber">5</span> <span class="toctext">Properties</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#Table_of_common_Z-transform_pairs"><span class="tocnumber">6</span> <span class="toctext">Table of common Z-transform pairs</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#Relationship_to_Laplace_transform"><span class="tocnumber">7</span> <span class="toctext">Relationship to Laplace transform</span></a></li>
<li class="toclevel-1 tocsection-15"><a href="#Relationship_to_Fourier_transform"><span class="tocnumber">8</span> <span class="toctext">Relationship to Fourier transform</span></a></li>
<li class="toclevel-1 tocsection-16"><a href="#Linear_constant-coefficient_difference_equation"><span class="tocnumber">9</span> <span class="toctext">Linear constant-coefficient difference equation</span></a>
<ul>
<li class="toclevel-2 tocsection-17"><a href="#Transfer_function"><span class="tocnumber">9.1</span> <span class="toctext">Transfer function</span></a></li>
<li class="toclevel-2 tocsection-18"><a href="#Zeros_and_poles"><span class="tocnumber">9.2</span> <span class="toctext">Zeros and poles</span></a></li>
<li class="toclevel-2 tocsection-19"><a href="#Output_response"><span class="tocnumber">9.3</span> <span class="toctext">Output response</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-20"><a href="#See_also"><span class="tocnumber">10</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-21"><a href="#References"><span class="tocnumber">11</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-22"><a href="#Further_reading"><span class="tocnumber">12</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-23"><a href="#External_links"><span class="tocnumber">13</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=1" title="Edit section: History">edit</a>]</span> <span class="mw-headline" id="History">History</span></h2>
<p>The basic idea now known as the Z-transform was known to <a href="http://en.wikipedia.org/wiki/Laplace" title="Laplace" class="mw-redirect">Laplace</a>, and re-introduced in 1947 by <a href="http://en.wikipedia.org/wiki/Witold_Hurewicz" title="Witold Hurewicz">W. Hurewicz</a> as a tractable way to solve linear, constant-coefficient <a href="http://en.wikipedia.org/wiki/Difference_equation" title="Difference equation" class="mw-redirect">difference equations</a>.<sup id="cite_ref-0" class="reference"><a href="#cite_note-0"><span>[</span>1<span>]</span></a></sup> It was later dubbed "the z-transform" by <a href="http://en.wikipedia.org/wiki/John_R._Ragazzini" title="John R. Ragazzini">Ragazzini</a> and <a href="http://en.wikipedia.org/wiki/Lotfi_A._Zadeh" title="Lotfi A. Zadeh">Zadeh</a> in the sampled-data control group at Columbia University in 1952.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>3<span>]</span></a></sup></p>
<p>The modified or <a href="http://en.wikipedia.org/wiki/Advanced_Z-transform" title="Advanced Z-transform">advanced Z-transform</a> was later developed and popularized by <a href="http://en.wikipedia.org/wiki/Eliahu_I._Jury" title="Eliahu I. Jury">E. I. Jury</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>4<span>]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>5<span>]</span></a></sup></p>
<p>The idea contained within the Z-transform is also known in mathematical literature as the method of <a href="http://en.wikipedia.org/wiki/Generating_function" title="Generating function">generating functions</a> which can be traced back as early as 1730 when it was introduced by <a href="http://en.wikipedia.org/wiki/Abraham_de_Moivre" title="Abraham de Moivre">de Moivre</a> in conjunction with probability theory.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>6<span>]</span></a></sup> From a mathematical view the Z-transform can also be viewed as a <a href="http://en.wikipedia.org/wiki/Laurent_series" title="Laurent series">Laurent series</a> where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=2" title="Edit section: Definition">edit</a>]</span> <span class="mw-headline" id="Definition">Definition</span></h2>
<p>The Z-transform, like many <a href="http://en.wikipedia.org/wiki/Integral_transforms" title="Integral transforms" class="mw-redirect">integral transforms</a>, can be defined as either a <i>one-sided</i> or <i>two-sided</i> transform.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=3" title="Edit section: Bilateral Z-transform">edit</a>]</span> <span class="mw-headline" id="Bilateral_Z-transform">Bilateral Z-transform</span></h3>
<p>The <i>bilateral</i> or <i>two-sided</i> Z-transform of a discrete-time signal <i>x[n]</i> is the <a href="http://en.wikipedia.org/wiki/Formal_power_series" title="Formal power series">formal power series</a> <i>X(z)</i> defined as</p>
<dl>
<dd><img class="tex" alt="X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} " src="Z-transform_pliki/4799f830eb930ecb74fe5368df6d7ab6.png"></dd>
</dl>
<p>where <i>n</i> is an integer and <i>z</i> is, in general, a <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex number</a>:</p>
<dl>
<dd><img class="tex" alt="z = A e^{j\phi} = A(\cos{\phi}+j\sin{\phi})\," src="Z-transform_pliki/1bdb462bb1b616f14bcc2d38a16181b3.png"></dd>
</dl>
<p>where <i>A</i> is the magnitude of <i>z</i>, <i>j</i> is the <a href="http://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, and <i><img class="tex" alt="\phi" src="Z-transform_pliki/7f20aa0b3691b496aec21cf356f63e04.png"></i> is the <i><a href="http://en.wikipedia.org/wiki/Complex_argument" title="Complex argument" class="mw-redirect">complex argument</a></i> (also referred to as <i>angle</i> or <i>phase</i>) in <a href="http://en.wikipedia.org/wiki/Radians" title="Radians" class="mw-redirect">radians</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=4" title="Edit section: Unilateral Z-transform">edit</a>]</span> <span class="mw-headline" id="Unilateral_Z-transform">Unilateral Z-transform</span></h3>
<p>Alternatively, in cases where <i>x</i>[<i>n</i>] is defined only for <i>n</i> ≥ 0, the <i>single-sided</i> or <i>unilateral</i> Z-transform is defined as</p>
<dl>
<dd><img class="tex" alt="X(z) = \mathcal{Z}\{x[n]\} =  \sum_{n=0}^{\infty} x[n] z^{-n}. \ " src="Z-transform_pliki/e342b69a450fef8889304628209e08c3.png"></dd>
</dl>
<p>In <a href="http://en.wikipedia.org/wiki/Signal_processing" title="Signal processing">signal processing</a>, this definition can be used to evaluate the Z-transform of the <a href="http://en.wikipedia.org/wiki/Finite_impulse_response#Impulse_response" title="Finite impulse response">unit impulse response</a> of a discrete-time <a href="http://en.wikipedia.org/wiki/Causal_system" title="Causal system">causal system</a>.</p>
<p>An important example of the unilateral Z-transform is the <a href="http://en.wikipedia.org/wiki/Probability-generating_function" title="Probability-generating function">probability-generating function</a>, where the component <img class="tex" alt="x[n]" src="Z-transform_pliki/d3baaa3204e2a03ef9528a7d631a4806.png"> is the probability that a discrete random variable takes the value <img class="tex" alt="n" src="Z-transform_pliki/7b8b965ad4bca0e41ab51de7b31363a1.png">, and the function <img class="tex" alt="X(z)" src="Z-transform_pliki/832dec27d15f24afcf0918b445133ca4.png"> is usually written as <img class="tex" alt="X(s)" src="Z-transform_pliki/280977f5eb0a373a76facb445b5166ee.png">, in terms of <img class="tex" alt="s = z^{-1}" src="Z-transform_pliki/3e89c30872f7e3d654b967c24ca3255c.png">. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=5" title="Edit section: Geophysical definition">edit</a>]</span> <span class="mw-headline" id="Geophysical_definition">Geophysical definition</span></h3>
<p>In geophysics, the usual definition for the Z-transform is a power series in <i>z</i> as opposed to <img class="tex" alt="z^{-1}" src="Z-transform_pliki/918e4ccac77d46420cac9652680b6b84.png">. This convention is used by Robinson and Treitel and by Kanasewich.<sup class="Template-Fact" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from May 2011">citation needed</span></a></i>]</sup> The geophysical definition is</p>
<dl>
<dd><img class="tex" alt="X(z) = \mathcal{Z}\{x[n]\} =  \sum_{n} x[n] z^{n}. \ " src="Z-transform_pliki/7adafd11922f7dc708923f753a1b470b.png"></dd>
</dl>
<p>The two definitions are equivalent; however, the difference results 
in a number of changes. For example, the location of zeros and poles 
move from inside the <a href="http://en.wikipedia.org/wiki/Unit_circle" title="Unit circle">unit circle</a>
 using one definition, to outside the unit circle using the other 
definition. Thus, care is required to note which definition is being 
used by a particular author.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=6" title="Edit section: Inverse Z-transform">edit</a>]</span> <span class="mw-headline" id="Inverse_Z-transform">Inverse Z-transform</span></h2>
<p>The <i>inverse</i> Z-transform is</p>
<dl>
<dd><img class="tex" alt=" x[n] = \mathcal{Z}^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz \ " src="Z-transform_pliki/06691f5e61f96167aed6ab2d28fa4363.png"></dd>
</dl>
<p>where <img class="tex" alt=" C \ " src="Z-transform_pliki/3161976361b830118b3b73fd24f09ba1.png"> is a counterclockwise closed path encircling the origin and entirely in the <a href="http://en.wikipedia.org/wiki/Radius_of_convergence" title="Radius of convergence">region of convergence</a> (ROC). The contour or path, <img class="tex" alt=" C \ " src="Z-transform_pliki/3161976361b830118b3b73fd24f09ba1.png">, must encircle all of the poles of <img class="tex" alt=" X(z) \ " src="Z-transform_pliki/985f5e20a8ff19d9596237e1c32f9561.png">.</p>
<p>A special case of this <a href="http://en.wikipedia.org/wiki/Contour_integral" title="Contour integral" class="mw-redirect">contour integral</a> occurs when <img class="tex" alt=" C \ " src="Z-transform_pliki/3161976361b830118b3b73fd24f09ba1.png"> is the unit circle (and can be used when the ROC includes the unit circle which is always guaranteed when <img class="tex" alt="X(z)\ " src="Z-transform_pliki/985f5e20a8ff19d9596237e1c32f9561.png"> is stable, i.e. all the poles are within the unit circle). The inverse Z-transform simplifies to the <a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform#Inverse_transform" title="Discrete-time Fourier transform">inverse discrete-time Fourier transform</a>:</p>
<dl>
<dd><img class="tex" alt=" x[n] = \frac{1}{2 \pi} \int_{-\pi}^{+\pi}  X(e^{j \omega}) e^{j \omega n} d \omega. \ " src="Z-transform_pliki/0835788bfa1380f6c0bc606f2da0a921.png"></dd>
</dl>
<p>The Z-transform with a finite range of <i>n</i> and a finite number of uniformly-spaced <i>z</i> values can be computed efficiently via <a href="http://en.wikipedia.org/wiki/Bluestein%27s_FFT_algorithm" title="Bluestein's FFT algorithm">Bluestein's FFT algorithm</a>. The <a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (DTFT) (not to be confused with the <a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> (DFT)) is a special case of such a Z-transform obtained by restricting <i>z</i> to lie on the unit circle.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=7" title="Edit section: Region of convergence">edit</a>]</span> <span class="mw-headline" id="Region_of_convergence">Region of convergence</span></h2>
<p>The <a href="http://en.wikipedia.org/wiki/Radius_of_convergence" title="Radius of convergence">region of convergence</a> (ROC) is the set of points in the complex plane for which the Z-transform summation converges.</p>
<dl>
<dd><img class="tex" alt="ROC = \left\{ z&nbsp;: \left|\sum_{n=-\infty}^{\infty}x[n]z^{-n}\right| &lt; \infty \right\} " src="Z-transform_pliki/96caf8885ff1d0531bed5670622fc90a.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=8" title="Edit section: Example 1 (no ROC)">edit</a>]</span> <span class="mw-headline" id="Example_1_.28no_ROC.29">Example 1 (no ROC)</span></h3>
<p>Let <img class="tex" alt="x[n] = 0.5^n\ " src="Z-transform_pliki/ae38b507e5e88ead1c7b0e035cde9046.png">. Expanding <img class="tex" alt="x[n]\ " src="Z-transform_pliki/c1466b9927640af95f78274058d272d9.png"> on the interval <img class="tex" alt="(-\infty, \infty)\ " src="Z-transform_pliki/d9662a88c311ab527dcf8afac3cc20ba.png"> it becomes</p>
<dl>
<dd><img class="tex" alt="x[n] = \{..., 0.5^{-3}, 0.5^{-2}, 0.5^{-1}, 1, 0.5, 0.5^2, 0.5^3, ...\} = \{..., 2^3, 2^2, 2, 1, 0.5, 0.5^2, 0.5^3, ...\}\ ." src="Z-transform_pliki/1e5620a51449d4620bcde1523cfdf689.png"></dd>
</dl>
<p>Looking at the sum</p>
<dl>
<dd><img class="tex" alt="\sum_{n=-\infty}^{\infty}x[n]z^{-n} \rightarrow \infty\ ." src="Z-transform_pliki/91b45b00cb7129c677014ab6f52c3753.png"></dd>
</dl>
<p>Therefore, there are no values of <img class="tex" alt="z\ " src="Z-transform_pliki/02f20ced3d6cc526b4cbc04e5911a5c7.png"> that satisfy this condition.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=9" title="Edit section: Example 2 (causal ROC)">edit</a>]</span> <span class="mw-headline" id="Example_2_.28causal_ROC.29">Example 2 (causal ROC)</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:252px;"><a href="http://en.wikipedia.org/wiki/File:Region_of_convergence_0.5_causal.svg" class="image"><img alt="" src="Z-transform_pliki/250px-Region_of_convergence_0_002.png" class="thumbimage" width="250" height="250"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Region_of_convergence_0.5_causal.svg" class="internal" title="Enlarge"><img src="Z-transform_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
ROC shown in blue, the unit circle as a dotted grey circle (appears reddish to the eye) and the circle <img class="tex" alt="\left|z\right| = 0.5" src="Z-transform_pliki/9de5b399dcbba0d1adae68e6a9436dc3.png"> is shown as a dashed black circle</div>
</div>
</div>
<p>Let <img class="tex" alt="x[n] = 0.5^n u[n]\ " src="Z-transform_pliki/57674052f63a5b5b55d69d0b097e5695.png"> (where <img class="tex" alt="u" src="Z-transform_pliki/7b774effe4a349c6dd82ad4f4f21d34c.png"> is the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>). Expanding <img class="tex" alt="x[n]\ " src="Z-transform_pliki/c1466b9927640af95f78274058d272d9.png"> on the interval <img class="tex" alt="(-\infty, \infty)\ " src="Z-transform_pliki/d9662a88c311ab527dcf8afac3cc20ba.png"> it becomes</p>
<dl>
<dd><img class="tex" alt="x[n] = \{..., 0, 0, 0, 1, 0.5, 0.5^2, 0.5^3, ...\}.\ " src="Z-transform_pliki/87b3f8c3d6429d893a4e6022247cd16a.png"></dd>
</dl>
<p>Looking at the sum</p>
<dl>
<dd><img class="tex" alt="\sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{\infty}0.5^nz^{-n} = \sum_{n=0}^{\infty}\left(\frac{0.5}{z}\right)^n = \frac{1}{1 - 0.5z^{-1}}.\ " src="Z-transform_pliki/21cdd138c51cbb6f6456cc9f15489c29.png"></dd>
</dl>
<p>The last equality arises from the infinite <a href="http://en.wikipedia.org/wiki/Geometric_series" title="Geometric series">geometric series</a> and the equality only holds if <img class="tex" alt="\left|0.5 z^{-1}\right| &lt; 1\ " src="Z-transform_pliki/00598fa9eb080beaed19ae8c51416ad1.png"> which can be rewritten in terms of <img class="tex" alt="z\ " src="Z-transform_pliki/02f20ced3d6cc526b4cbc04e5911a5c7.png"> as <img class="tex" alt="\left|z\right| &gt; 0.5\ " src="Z-transform_pliki/9c965941bfc9e257f9de951e508e7098.png">. Thus, the ROC is <img class="tex" alt="\left|z\right| &gt; 0.5\ " src="Z-transform_pliki/9c965941bfc9e257f9de951e508e7098.png">. In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".<br clear="all"></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=10" title="Edit section: Example 3 (anticausal ROC)">edit</a>]</span> <span class="mw-headline" id="Example_3_.28anticausal_ROC.29">Example 3 (anticausal ROC)</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:252px;"><a href="http://en.wikipedia.org/wiki/File:Region_of_convergence_0.5_anticausal.svg" class="image"><img alt="" src="Z-transform_pliki/250px-Region_of_convergence_0_003.png" class="thumbimage" width="250" height="250"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Region_of_convergence_0.5_anticausal.svg" class="internal" title="Enlarge"><img src="Z-transform_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
ROC shown in blue, the unit circle as a dotted grey circle and the circle <img class="tex" alt="\left|z\right| = 0.5" src="Z-transform_pliki/9de5b399dcbba0d1adae68e6a9436dc3.png"> is shown as a dashed black circle</div>
</div>
</div>
<p>Let <img class="tex" alt="x[n] = -(0.5)^n u[-n-1]\ " src="Z-transform_pliki/9508c0a3dd6fab9c51bd316852844bf1.png"> (where <img class="tex" alt="u" src="Z-transform_pliki/7b774effe4a349c6dd82ad4f4f21d34c.png"> is the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>). Expanding <img class="tex" alt="x[n]\ " src="Z-transform_pliki/c1466b9927640af95f78274058d272d9.png"> on the interval <img class="tex" alt="(-\infty, \infty)\ " src="Z-transform_pliki/d9662a88c311ab527dcf8afac3cc20ba.png"> it becomes</p>
<dl>
<dd><img class="tex" alt="x[n] = \{..., -(0.5)^{-3}, -(0.5)^{-2}, -(0.5)^{-1}, 0, 0, 0, 0,  ...\}.\ " src="Z-transform_pliki/87908bb58989914865eebbc1776aa508.png"></dd>
</dl>
<p>Looking at the sum</p>
<dl>
<dd><img class="tex" alt="\sum_{n=-\infty}^{\infty}x[n]z^{-n} = -\sum_{n=-\infty}^{-1}0.5^nz^{-n} = -\sum_{n=-\infty}^{-1}\left(\frac{z}{0.5}\right)^{-n}\ " src="Z-transform_pliki/1e6a35a12a88d07372c071a528f1b5d1.png"></dd>
<dd><img class="tex" alt="= -\sum_{m=1}^{\infty}\left(\frac{z}{0.5}\right)^{m} = -\frac{0.5^{-1}z}{1 - 0.5^{-1}z} = \frac{z}{z - 0.5} = \frac{1}{1 - 0.5z^{-1}}.\ " src="Z-transform_pliki/84eaa2453bba2ae5c6f25425a62931cb.png"></dd>
</dl>
<p>Using the infinite <a href="http://en.wikipedia.org/wiki/Geometric_series" title="Geometric series">geometric series</a>, again, the equality only holds if <img class="tex" alt="\left|0.5^{-1}z\right| &lt; 1\ " src="Z-transform_pliki/dd25cb99de1441802393e972f77da6dd.png"> which can be rewritten in terms of <img class="tex" alt="z\ " src="Z-transform_pliki/02f20ced3d6cc526b4cbc04e5911a5c7.png"> as <img class="tex" alt="\left|z\right| &lt; 0.5\ " src="Z-transform_pliki/3cb67167bdb4ed99bb69323c43222b45.png">. Thus, the ROC is <img class="tex" alt="\left|z\right| &lt; 0.5\ " src="Z-transform_pliki/3cb67167bdb4ed99bb69323c43222b45.png">. In this case the ROC is a disc centered at the origin and of radius 0.5.</p>
<p>What differentiates this example from the previous example is <i>only</i> the ROC. This is intentional to demonstrate that the transform result alone is insufficient.<br clear="all"></p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=11" title="Edit section: Examples conclusion">edit</a>]</span> <span class="mw-headline" id="Examples_conclusion">Examples conclusion</span></h3>
<p>Examples 2 &amp; 3 clearly show that the Z-transform <img class="tex" alt="X(z)\ " src="Z-transform_pliki/985f5e20a8ff19d9596237e1c32f9561.png"> of <img class="tex" alt="x[n]\ " src="Z-transform_pliki/c1466b9927640af95f78274058d272d9.png"> is unique when and only when specifying the ROC. Creating the <a href="http://en.wikipedia.org/wiki/Pole-zero_plot" title="Pole-zero plot" class="mw-redirect">pole-zero plot</a>
 for the causal and anticausal case show that the ROC for either case 
does not include the pole that is at 0.5. This extends to cases with 
multiple poles: the ROC will <i>never</i> contain poles.</p>
<p>In example 2, the causal system yields an ROC that includes <img class="tex" alt="\left| z \right| = \infty\ " src="Z-transform_pliki/55cc85473bda18ca19f1101b8772948e.png"> while the anticausal system in example 3 yields an ROC that includes <img class="tex" alt="\left| z \right| = 0\ " src="Z-transform_pliki/7226151c71ac5c484e3df2bc5c804663.png">.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:252px;"><a href="http://en.wikipedia.org/wiki/File:Region_of_convergence_0.5_0.75_mixed-causal.svg" class="image"><img alt="" src="Z-transform_pliki/250px-Region_of_convergence_0.png" class="thumbimage" width="250" height="250"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Region_of_convergence_0.5_0.75_mixed-causal.svg" class="internal" title="Enlarge"><img src="Z-transform_pliki/magnify-clip.png" alt="" width="15" height="11"></a></div>
ROC shown as a blue ring <img class="tex" alt="0.5 &lt; \left| z \right| &lt; 0.75\ " src="Z-transform_pliki/c4179ff1fb2a099a32621a94c3af30db.png"></div>
</div>
</div>
<p>In systems with multiple poles it is possible to have an ROC that includes neither <img class="tex" alt="\left| z \right| = \infty\ " src="Z-transform_pliki/55cc85473bda18ca19f1101b8772948e.png"> nor <img class="tex" alt="\left| z \right| = 0\ " src="Z-transform_pliki/7226151c71ac5c484e3df2bc5c804663.png">. The ROC creates a circular band. For example, <img class="tex" alt="x[n] = 0.5^nu[n] - 0.75^nu[-n-1]\ " src="Z-transform_pliki/5bb87341b337d0d121453a4b16d01d31.png"> has poles at 0.5 and 0.75. The ROC will be <img class="tex" alt="0.5 &lt; \left| z \right| &lt; 0.75\ " src="Z-transform_pliki/c4179ff1fb2a099a32621a94c3af30db.png">, which includes neither the origin nor infinity. Such a system is called a <a href="http://en.wikipedia.org/w/index.php?title=Mixed-causality_system&amp;action=edit&amp;redlink=1" class="new" title="Mixed-causality system (page does not exist)">mixed-causality system</a> as it contains a causal term <img class="tex" alt="0.5^nu[n]\ " src="Z-transform_pliki/3133def49d00cf59a1e73e28b6988e9c.png"> and an anticausal term <img class="tex" alt="-(0.75)^nu[-n-1]\ " src="Z-transform_pliki/6c39198fd187e5e3a8b2993efc0afceb.png">.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Control_theory#Stability" title="Control theory">stability</a> of a system can also be determined by knowing the ROC alone. If the ROC contains the unit circle (i.e., <img class="tex" alt="\left| z \right| = 1\ " src="Z-transform_pliki/8bf9b411c9ceb175b8b6ba8d427c6a67.png">) then the system is stable. In the above systems the causal system (Example 2) is stable because <img class="tex" alt="\left| z \right| &gt; 0.5\ " src="Z-transform_pliki/9c965941bfc9e257f9de951e508e7098.png"> contains the unit circle.</p>
<p>If you are provided a Z-transform of a system without an ROC (i.e., an ambiguous <img class="tex" alt="x[n]\ " src="Z-transform_pliki/c1466b9927640af95f78274058d272d9.png">) you can determine a unique <img class="tex" alt="x[n]\ " src="Z-transform_pliki/c1466b9927640af95f78274058d272d9.png"> provided you desire the following:</p>
<ul>
<li>Stability</li>
<li>Causality</li>
</ul>
<p>If you need stability then the ROC must contain the unit circle. If 
you need a causal system then the ROC must contain infinity and the 
system function will be a right-sided sequence. If you need an 
anticausal system then the ROC must contain the origin and the system 
function will be a left-sided sequence. If you need both, stability and 
causality, all the poles of the system function must be inside the unit 
circle.</p>
<p>The unique <img class="tex" alt="x[n]\ " src="Z-transform_pliki/c1466b9927640af95f78274058d272d9.png"> can then be found.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=12" title="Edit section: Properties">edit</a>]</span> <span class="mw-headline" id="Properties">Properties</span></h2>
<table class="wikitable">
<caption><b>Properties of the z-transform</b></caption>
<tbody><tr>
<th></th>
<th>Time domain</th>
<th>Z-domain</th>
<th>Proof</th>
<th>ROC</th>
</tr>
<tr>
<th>Notation</th>
<td><img class="tex" alt="x[n]=\mathcal{Z}^{-1}\{X(z)\}" src="Z-transform_pliki/ac2e4e0332cbbe3296dd601c898b14cf.png"></td>
<td><img class="tex" alt="X(z)=\mathcal{Z}\{x[n]\}" src="Z-transform_pliki/949157e85e0059c0ef07f00c2c26bc2e.png"></td>
<td></td>
<td>ROC: <img class="tex" alt="r_2&lt;|z|&lt;r_1 \ " src="Z-transform_pliki/2f6b913e430abfc243aaf0393574d07f.png"></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Linearity" title="Linearity">Linearity</a></th>
<td><img class="tex" alt="a_1 x_1[n] + a_2 x_2[n]\ " src="Z-transform_pliki/79bdec3451b8f328c4460e35b821b8ea.png"></td>
<td><img class="tex" alt="a_1 X_1(z) + a_2 X_2(z) \ " src="Z-transform_pliki/67e35ed8e26c86db82e16ad7bf82825e.png"></td>
<td><img class="tex" alt="\begin{array} {lcl} X(z) = &amp; \\
         \sum_{n=-\infty}^{\infty} (a_1x_1(n)+a_2x_2(n))z^{-n}\ &amp; \\
         = a_1\sum_{n=-\infty}^{\infty} (x_1(n))z^{-n} + &amp; \\
a_2\sum_{n=-\infty}^{\infty}(x_2(n))z^{-n} &amp; \\
         = a_1X_1(z) + a_2X_2(z)\end{array} " src="Z-transform_pliki/4414763fe35b01e2a8efc5d03ccfd0ee.png"></td>
<td>At least the intersection of ROC<sub>1</sub> and ROC<sub>2</sub></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/w/index.php?title=Time_expansion&amp;action=edit&amp;redlink=1" class="new" title="Time expansion (page does not exist)">Time expansion</a></th>
<td><img class="tex" alt="x_{(k)}[n] = \begin{cases} x[r], &amp; n = rk \\ 0, &amp; n \not= rk \end{cases}" src="Z-transform_pliki/48f5fea0a96a642423ad90e209a52e6b.png">
<p><img class="tex" alt="r" src="Z-transform_pliki/4b43b0aee35624cd95b910189b3dc231.png">: integer</p>
</td>
<td><img class="tex" alt="X(z^k) \ " src="Z-transform_pliki/890fd2aa778b912160b9a8e6cd73d490.png"></td>
<td></td>
<td>R^{1/k}</td>
</tr>
<tr>
<th>Time shifting</th>
<td><img class="tex" alt="x[n-k]\ " src="Z-transform_pliki/eb62e81664b3d3f149e9c52f2df4a09b.png"></td>
<td><img class="tex" alt="z^{-k}X(z) \ " src="Z-transform_pliki/534ecdbd3e16bb9b9869695b84e2c453.png"></td>
<td><img class="tex" alt=" \begin{array} {lcl} Z\{x[n-k]\} = &amp; \\ 
\sum_{n=0}^{\infty} x[n-k]z^{-n}\&amp; \\
\text{    ,let }j = n - k &amp; \\
= \sum_{j=-k}^{\infty} x[j]z^{-(j+k)}&amp; \\
= \sum_{j=-k}^{\infty} x[j]z^{-j}z^{-k}&amp; \\
= z^{-k}\sum_{j=-k}^{\infty}x[j]z^{-j}&amp; \\
= z^{-k}\sum_{j=0}^{\infty}x[j]z^{-j} &amp; \\
\text{    , since }x[\beta]=0 \text{ if }\beta&lt;0 &amp; \\
= z^{-k}X(z)&amp; \\
\end{array} " src="Z-transform_pliki/e7e953cb75cfb298dafbc6c8453798ad.png"></td>
<td>ROC, except <img class="tex" alt="z=0\ " src="Z-transform_pliki/59c04b45711dcc0858511481602abe1b.png"> if <img class="tex" alt="k&gt;0\," src="Z-transform_pliki/04d4a4b969a9937e007085d733918c7f.png"> and <img class="tex" alt="z=\infty" src="Z-transform_pliki/6fb3d985fc0f3e3d2ea869467a29df33.png"> if <img class="tex" alt="k&lt;0\ " src="Z-transform_pliki/2e867f2bbe2f1da5545bdf395674752d.png"></td>
</tr>
<tr>
<th>Scaling in
<p>the z-domain</p>
</th>
<td><img class="tex" alt="a^n x[n]\ " src="Z-transform_pliki/5bc0cde4af5f3bb27463af9a62dc9a20.png"></td>
<td><img class="tex" alt="X(a^{-1}z) \ " src="Z-transform_pliki/f3660436967d50bb19921d973d560be9.png"></td>
<td><img class="tex" alt="\begin{array} {lcl} Z \{a^n x[n]\} = &amp; \\
\sum_{n=-\infty}^{\infty} a^{n}x(n)z^{-n}&amp; \\
= \sum_{n=-\infty}^{\infty} x(n)(a^{-1}z)^{-n} &amp; \\
= X(a^{-1}z) &amp; \\
\end{array} " src="Z-transform_pliki/ede1dd8d8e484b39864d125bca60f898.png"></td>
<td><img class="tex" alt="|a|r_2&lt;|z|&lt;|a|r_1 \ " src="Z-transform_pliki/0649190b7966c7a19278df893c8ad9ca.png"></td>
</tr>
<tr>
<th>Time reversal</th>
<td><img class="tex" alt="x[-n]\ " src="Z-transform_pliki/daffc5bc57063f1d1bc403656e2d6096.png"></td>
<td><img class="tex" alt="X(z^{-1}) \ " src="Z-transform_pliki/299336010ae306d6e93992d73a1c8942.png"></td>
<td><img class="tex" alt="\begin{array} {lcl} \mathcal{Z}\{x(-n)\} = &amp; \\ 
\sum_{n=-\infty}^{\infty} x(-n)z^{-n}\ &amp; \\
= \sum_{m=-\infty}^{\infty} x(m)z^{m}\ &amp; \\
= \sum_{m=-\infty}^{\infty} x(m){(z^{-1})}^{-m}\ &amp; \\
= X(z^{-1}) &amp; \\
\end{array} " src="Z-transform_pliki/7ccd9244db4806e53583397e7d651a02.png"></td>
<td><img class="tex" alt="\frac{1}{r_1}&lt;|z|&lt;\frac{1}{r_2} \ " src="Z-transform_pliki/a51b55a3e068fb5a176298cf88ddcf5d.png"></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Complex_conjugation" title="Complex conjugation" class="mw-redirect">Complex conjugation</a></th>
<td><img class="tex" alt="x^*[n]\ " src="Z-transform_pliki/af74a1c4213dafcf6395eeb45e9ac8d2.png"></td>
<td><img class="tex" alt="X^*(z^*) \ " src="Z-transform_pliki/4bba41ee07b61e27a6f49e78321615c7.png"></td>
<td><img class="tex" alt="\begin{array} {lcl}Z\{x^*(n)\} = &amp; \\ 
\sum_{n=-\infty}^{\infty} x^*(n)z^{-n}\ &amp; \\
= \sum_{n=-\infty}^{\infty} [x(n)(z^*)^{-n}]^*\ &amp; \\
= [ \sum_{n=-\infty}^{\infty} x(n)(z^*)^{-n}\ ]^* &amp; \\
= X^*(z^*)&amp; \\
\end{array} " src="Z-transform_pliki/0fdf7d718324c524316a65ae63c8e406.png"></td>
<td>ROC</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Real_part" title="Real part" class="mw-redirect">Real part</a></th>
<td><img class="tex" alt="\operatorname{Re}\{x[n]\}\ " src="Z-transform_pliki/7fa84bf49517a5bdb652bcfd0f8a7419.png"></td>
<td><img class="tex" alt="\frac{1}{2}\left[X(z)+X^*(z^*) \right]" src="Z-transform_pliki/f11b75b76a9c8fbf3b90f47799e92b92.png"></td>
<td></td>
<td>ROC</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Imaginary_part" title="Imaginary part" class="mw-redirect">Imaginary part</a></th>
<td><img class="tex" alt="\operatorname{Im}\{x[n]\}\ " src="Z-transform_pliki/9d79da92f4c39771f2fc49371473527a.png"></td>
<td><img class="tex" alt="\frac{1}{2j}\left[X(z)-X^*(z^*) \right]" src="Z-transform_pliki/b3e01ee80092b499fcdeeed27f5f358d.png"></td>
<td></td>
<td>ROC</td>
</tr>
<tr>
<th>Differentiation</th>
<td><img class="tex" alt="nx[n]\ " src="Z-transform_pliki/d8b2e4a3a8c01427adf3b5a3159cd6aa.png"></td>
<td><img class="tex" alt=" -z \frac{dX(z)}{dz}" src="Z-transform_pliki/1991391dd9b9a6915b626d3c6d4836be.png"></td>
<td><img class="tex" alt="\begin{array} {lcl}Z\{nx(n)\} = &amp; \\   
\sum_{n=-\infty}^{\infty} nx(n)z^{-n}\  &amp; \\
= z  \sum_{n=-\infty}^{\infty} nx(n)z^{-n-1}\ &amp; \\
= -z  \sum_{n=-\infty}^{\infty} x(n)(-nz^{-n-1})\ &amp; \\
= -z  \sum_{n=-\infty}^{\infty} x(n)\frac{d}{dz}(z^{-n})\ &amp; \\
= -z \frac{dX(z)}{dz}&amp; \\
\end{array} " src="Z-transform_pliki/00efc54574901e8eb7f5b331a5ec6c37.png"></td>
<td>ROC</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">Convolution</a></th>
<td><img class="tex" alt="x_1[n] * x_2[n]\ " src="Z-transform_pliki/84306084618dd955215d355316dbd034.png"></td>
<td><img class="tex" alt="X_1(z)X_2(z) \ " src="Z-transform_pliki/fd6a1bae5a202e4d423b329092d561ef.png"></td>
<td><img class="tex" alt="\begin{array} {lcl}\mathcal{Z}\{x_1(n)*x_2(n)\} = &amp; \\
                                   \mathcal{Z} \{\sum_{l=-\infty}^{\infty} x_1(l)x_2(n-l)\}\ &amp; \\
                                   = \sum_{n=-\infty}^{\infty} [\sum_{l=-\infty}^{\infty} x_1(l)x_2(n-l)]z^{-n}\ &amp; \\
                                   =\sum_{l=-\infty}^{\infty} x_1(l) \sum_{n=-\infty}^{\infty} x_2(n-l)z^{-n} ]\ &amp; \\
                                   =[\sum_{l=-\infty}^{\infty} x_1(l)z^{-l}] [\sum_{n=-\infty}^{\infty} x_2(n)z^{-n} ]\ &amp; \\
                                   =X_1(z)X_2(z)&amp; \\
\end{array} " src="Z-transform_pliki/296054afa7c97e46c5c20fb57895caa6.png"></td>
<td>At least the intersection of ROC<sub>1</sub> and ROC<sub>2</sub></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation</a></th>
<td><img class="tex" alt="r_{x_1,x_2}=x_1^*[-n] * x_2[n]\ " src="Z-transform_pliki/ce490b2de558bd99471866615bdd5b09.png"></td>
<td><img class="tex" alt="R_{x_1,x_2}(z)=X_1^*(1/z^*)X_2(z)\ " src="Z-transform_pliki/58fefc4e983b8f1cb367eac4d28d14a1.png"></td>
<td></td>
<td>At least the intersection of ROC of <img class="tex" alt="X_1(1/z^*)" src="Z-transform_pliki/7d10439cc8ca7f1a4fbe6e536a36a3f5.png"> and <img class="tex" alt="X_2(z)" src="Z-transform_pliki/81e47f700360bbf877a0f54287823814.png"></td>
</tr>
<tr>
<th>First difference</th>
<td><img class="tex" alt="x[n] - x[n-1] \ " src="Z-transform_pliki/aa6030ba7dce7f2b8e7fb99082042066.png"></td>
<td><img class="tex" alt=" (1-z^{-1})X(z) \ " src="Z-transform_pliki/1de427a0060f0e468303cb330467713b.png"></td>
<td></td>
<td>At least the intersection of ROC of X<sub>1</sub>(z) and <img class="tex" alt="|z|&gt;0" src="Z-transform_pliki/e4219b586ee9819e560a79e39978613b.png"></td>
</tr>
<tr>
<th>Accumulation</th>
<td><img class="tex" alt="\sum_{k=-\infty}^{n} x[k]\ " src="Z-transform_pliki/5f3de91aaf3bbc0a76508a92c7b4633a.png"></td>
<td><img class="tex" alt=" \frac{1}{1-z^{-1} }X(z)" src="Z-transform_pliki/9efd2f5accc6da7708a31cb35151727e.png"></td>
<td><img class="tex" alt="\begin{array} {lcl}\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{n} x[k]\cdot z^{-n}\\
        =\sum_{n=-\infty}^{\infty}(x[n]+x[n-1]+\\
x[n-2]\cdots x[-\infty])z^{-n}\\
        =X[z](1+z^{-1}+z^{-2}+z^{-3}\cdots )\\
        =X[z]\sum_{j=0}^{\infty}z^{-j}   \\
        =X[z] \frac{1}{1-z^{-1}}\end{array}" src="Z-transform_pliki/ba01ec6eb89fea619c76096b990de53c.png"></td>
<td></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication">Multiplication</a></th>
<td><img class="tex" alt="x_1[n]x_2[n]\ " src="Z-transform_pliki/bd176884c506fcef3604fd6d23de884b.png"></td>
<td><img class="tex" alt="\frac{1}{j2\pi}\oint_C X_1(v)X_2(\frac{z}{v})v^{-1}\mathrm{d}v \ " src="Z-transform_pliki/e125e574850eb3de9b4b3c4760ffa1bf.png"></td>
<td></td>
<td class="tex">-</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's relation</a></th>
<td><img class="tex" alt="\sum_{n=-\infty}^{\infty} x_1[n]x^*_2[n]\ " src="Z-transform_pliki/7c33541ee039439ee2a60d83ad93a0dd.png"></td>
<td><img class="tex" alt="\frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\frac{1}{v^*})v^{-1}\mathrm{d}v \ " src="Z-transform_pliki/af71da63eca8c159bd93ce958aadf403.png"></td>
<td></td>
<td></td>
</tr>
</tbody></table>
<ul>
<li><b><a href="http://en.wikipedia.org/wiki/Initial_value_theorem" title="Initial value theorem">Initial value theorem</a></b></li>
</ul>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="x[0]=\lim_{z\rightarrow \infty}X(z) \ " src="Z-transform_pliki/92bb0bebca1839c58a88b6feff81c6aa.png">, If <img class="tex" alt="x[n]\," src="Z-transform_pliki/b0f955b87baf7377b98b47ed9c723949.png"> causal</dd>
</dl>
</dd>
</dl>
<ul>
<li><b><a href="http://en.wikipedia.org/wiki/Final_value_theorem" title="Final value theorem">Final value theorem</a></b></li>
</ul>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="x[\infty]=\lim_{z\rightarrow 1}(z-1)X(z) \ " src="Z-transform_pliki/79f8f3555de2eba7269fee2a2ca98dc9.png">, Only if poles of <img class="tex" alt="(z-1)X(z) \ " src="Z-transform_pliki/b1b8bd57acba0648eb0c6fc4a121119a.png"> are inside the unit circle</dd>
</dl>
</dd>
</dl>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=13" title="Edit section: Table of common Z-transform pairs">edit</a>]</span> <span class="mw-headline" id="Table_of_common_Z-transform_pairs">Table of common Z-transform pairs</span></h2>
<p>Here:</p>
<ul>
<li><img class="tex" alt="u[n] = \begin{cases} 1, &amp; n \ge 0 \\ 0, &amp; n &lt; 0 \end{cases}" src="Z-transform_pliki/ebaeed1cdf2a16626315fbb5684b3552.png"></li>
<li><img class="tex" alt="\delta[n] = \begin{cases} 1, &amp; n = 0 \\ 0, &amp; n \ne 0 \end{cases}" src="Z-transform_pliki/af1026c1cbf08b31325071a939cf0c07.png"></li>
</ul>
<table class="wikitable">
<tbody><tr>
<th></th>
<th>Signal, <img class="tex" alt="x[n]" src="Z-transform_pliki/d3baaa3204e2a03ef9528a7d631a4806.png"></th>
<th>Z-transform, <img class="tex" alt="X(z)" src="Z-transform_pliki/832dec27d15f24afcf0918b445133ca4.png"></th>
<th>ROC</th>
</tr>
<tr>
<td>1</td>
<td><img class="tex" alt="\delta[n] \, " src="Z-transform_pliki/2b63622fadf95b2200b264909054224f.png"></td>
<td><img class="tex" alt="1\, " src="Z-transform_pliki/d06c48671eacd7f1e2afde7289e483d5.png"></td>
<td><img class="tex" alt=" \mbox{all }z\, " src="Z-transform_pliki/8526e84d7c17e402d225950ad29139b0.png"></td>
</tr>
<tr>
<td>2</td>
<td><img class="tex" alt="\delta[n-n_0] \," src="Z-transform_pliki/4c035051ef51cb09d5cbe903b496208a.png"></td>
<td><img class="tex" alt=" z^{-n_0} \, " src="Z-transform_pliki/bb59029e4d141a40804cdcfb243853ee.png"></td>
<td><img class="tex" alt=" z \neq 0\," src="Z-transform_pliki/539443b3ade242577794742e2115904e.png"></td>
</tr>
<tr>
<td>3</td>
<td><img class="tex" alt="u[n] \," src="Z-transform_pliki/7016daf9693a54fbb365146aa38d73c6.png"></td>
<td><img class="tex" alt=" \frac{1}{1-z^{-1} }" src="Z-transform_pliki/e735b90245eacb5cc3b0458e248f5f93.png"></td>
<td><img class="tex" alt="|z| &gt; 1\," src="Z-transform_pliki/b3e5447ee5136c29e096f5d260c3a830.png"></td>
</tr>
<tr>
<td>4</td>
<td><img class="tex" alt="\, e^{-\alpha n} u[n]  " src="Z-transform_pliki/35097e6a8b51f48e543bf37957ed6d68.png"></td>
<td><img class="tex" alt="  1 \over 1-e^{-\alpha  }z^{-1}" src="Z-transform_pliki/effd80c89b739d6ba1ed2033774a18fe.png"></td>
<td><img class="tex" alt="  |z| &gt;  |e^{-\alpha}| \," src="Z-transform_pliki/f5b448128a3c16f41bfd96c77805c4fc.png"></td>
</tr>
<tr>
<td>5</td>
<td><img class="tex" alt=" - u[-n-1] \," src="Z-transform_pliki/596e922d21a3ca551fb1805ce332759e.png"></td>
<td><img class="tex" alt=" \frac{1}{1 - z^{-1}}" src="Z-transform_pliki/e735b90245eacb5cc3b0458e248f5f93.png"></td>
<td><img class="tex" alt="|z| &lt; 1\," src="Z-transform_pliki/db4601b2361acef6dd5780297bd49911.png"></td>
</tr>
<tr>
<td>6</td>
<td><img class="tex" alt=" n u[n] \," src="Z-transform_pliki/1654b58cc296812ba337d3753898834b.png"></td>
<td><img class="tex" alt=" \frac{z^{-1}}{( 1-z^{-1} )^2}" src="Z-transform_pliki/2317d8a1f881d94ddaaa4cbc7212d467.png"></td>
<td><img class="tex" alt="|z| &gt; 1\," src="Z-transform_pliki/b3e5447ee5136c29e096f5d260c3a830.png"></td>
</tr>
<tr>
<td>7</td>
<td><img class="tex" alt=" - n u[-n-1] \," src="Z-transform_pliki/41b866b5f12cc275d702937c3a929222.png"></td>
<td><img class="tex" alt=" \frac{z^{-1} }{ (1 - z^{-1})^2 }" src="Z-transform_pliki/2317d8a1f881d94ddaaa4cbc7212d467.png"></td>
<td><img class="tex" alt=" |z| &lt; 1 \," src="Z-transform_pliki/db4601b2361acef6dd5780297bd49911.png"></td>
</tr>
<tr>
<td>8</td>
<td><img class="tex" alt="n^2 u[n] \," src="Z-transform_pliki/3d24a549af9143a2482c7d169e135795.png"></td>
<td><img class="tex" alt="  \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} " src="Z-transform_pliki/9a1706340d0e65b104681df3a00e6a7e.png"></td>
<td><img class="tex" alt="|z| &gt; 1\," src="Z-transform_pliki/b3e5447ee5136c29e096f5d260c3a830.png"></td>
</tr>
<tr>
<td>9</td>
<td><img class="tex" alt=" - n^2 u[-n - 1] \," src="Z-transform_pliki/78282ba68d8f36a1586b3247cbdd5674.png"></td>
<td><img class="tex" alt="  \frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} " src="Z-transform_pliki/9a1706340d0e65b104681df3a00e6a7e.png"></td>
<td><img class="tex" alt="|z| &lt; 1\," src="Z-transform_pliki/db4601b2361acef6dd5780297bd49911.png"></td>
</tr>
<tr>
<td>10</td>
<td><img class="tex" alt="n^3 u[n] \," src="Z-transform_pliki/40044ac2551be5de950fc05a4fbcb30f.png"></td>
<td><img class="tex" alt=" \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} " src="Z-transform_pliki/3ad4f12f66da73b1d4bac12fd4de731d.png"></td>
<td><img class="tex" alt="|z| &gt; 1\," src="Z-transform_pliki/b3e5447ee5136c29e096f5d260c3a830.png"></td>
</tr>
<tr>
<td>11</td>
<td><img class="tex" alt="- n^3 u[-n -1] \," src="Z-transform_pliki/6f1d679d09c86f67ae88195f6307fde6.png"></td>
<td><img class="tex" alt=" \frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} " src="Z-transform_pliki/3ad4f12f66da73b1d4bac12fd4de731d.png"></td>
<td><img class="tex" alt="|z| &lt; 1\," src="Z-transform_pliki/db4601b2361acef6dd5780297bd49911.png"></td>
</tr>
<tr>
<td>12</td>
<td><img class="tex" alt="a^n u[n] \," src="Z-transform_pliki/52005e1c22b667a92f6a7f8763d198aa.png"></td>
<td><img class="tex" alt=" \frac{1}{1-a z^{-1}}" src="Z-transform_pliki/1206f45dc4438dd5f5b55e34d596f9a8.png"></td>
<td><img class="tex" alt=" |z| &gt; |a|\," src="Z-transform_pliki/2a693d4762bb2c8e1e4e11d6aa5ddce7.png"></td>
</tr>
<tr>
<td>13</td>
<td><img class="tex" alt="-a^n u[-n-1] \," src="Z-transform_pliki/5b1d6d741e4466bd975e49b8a7502a06.png"></td>
<td><img class="tex" alt=" \frac{1}{1-a z^{-1}}" src="Z-transform_pliki/1206f45dc4438dd5f5b55e34d596f9a8.png"></td>
<td><img class="tex" alt="|z| &lt; |a|\," src="Z-transform_pliki/933fdf052c34ef96668d8f4694188c5d.png"></td>
</tr>
<tr>
<td>14</td>
<td><img class="tex" alt="n a^n u[n] \," src="Z-transform_pliki/a5ee7e0b460ced4724323abe028b7d5f.png"></td>
<td><img class="tex" alt=" \frac{az^{-1} }{ (1-a z^{-1})^2 }" src="Z-transform_pliki/927bcbf827cb6383708443c86a7d9dea.png"></td>
<td><img class="tex" alt="|z| &gt; |a|\," src="Z-transform_pliki/2a693d4762bb2c8e1e4e11d6aa5ddce7.png"></td>
</tr>
<tr>
<td>15</td>
<td><img class="tex" alt="-n a^n u[-n-1] \," src="Z-transform_pliki/5422993372c0c804ccdc7c6d3f62c7b0.png"></td>
<td><img class="tex" alt=" \frac{az^{-1} }{ (1-a z^{-1})^2 }" src="Z-transform_pliki/927bcbf827cb6383708443c86a7d9dea.png"></td>
<td><img class="tex" alt=" |z| &lt; |a|\," src="Z-transform_pliki/933fdf052c34ef96668d8f4694188c5d.png"></td>
</tr>
<tr>
<td>16</td>
<td><img class="tex" alt="n^2 a^n u[n] \," src="Z-transform_pliki/5e752df8b1b2c1be5b169617d3d885e8.png"></td>
<td><img class="tex" alt=" \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} " src="Z-transform_pliki/70870a0557348519a64c8686f648406a.png"></td>
<td><img class="tex" alt="|z| &gt; |a|\," src="Z-transform_pliki/2a693d4762bb2c8e1e4e11d6aa5ddce7.png"></td>
</tr>
<tr>
<td>17</td>
<td><img class="tex" alt="- n^2 a^n u[-n -1] \," src="Z-transform_pliki/2af9e2fbcb9952df47812c829b6477d9.png"></td>
<td><img class="tex" alt=" \frac{a z^{-1} (1 + a z^{-1}) }{(1-a z^{-1})^3} " src="Z-transform_pliki/70870a0557348519a64c8686f648406a.png"></td>
<td><img class="tex" alt="|z| &lt; |a|\," src="Z-transform_pliki/933fdf052c34ef96668d8f4694188c5d.png"></td>
</tr>
<tr>
<td>18</td>
<td><img class="tex" alt="\cos(\omega_0 n) u[n] \," src="Z-transform_pliki/57a085c1d96479f7dd14f6f3d76e0520.png"></td>
<td><img class="tex" alt=" \frac{ 1-z^{-1} \cos(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }" src="Z-transform_pliki/1c8e515d81c817fc9a4c09b81a2d0daa.png"></td>
<td><img class="tex" alt=" |z| &gt;1\," src="Z-transform_pliki/b3e5447ee5136c29e096f5d260c3a830.png"></td>
</tr>
<tr>
<td>19</td>
<td><img class="tex" alt="\sin(\omega_0 n) u[n] \," src="Z-transform_pliki/15b15b84c75d60afebabb9fc0c8acb51.png"></td>
<td><img class="tex" alt=" \frac{ z^{-1} \sin(\omega_0) }{ 1-2z^{-1}\cos(\omega_0)+ z^{-2} }" src="Z-transform_pliki/c242cfe38471cc2153b4f1b8f1e88d74.png"></td>
<td><img class="tex" alt=" |z| &gt;1\," src="Z-transform_pliki/b3e5447ee5136c29e096f5d260c3a830.png"></td>
</tr>
<tr>
<td>20</td>
<td><img class="tex" alt="a^n \cos(\omega_0 n) u[n] \," src="Z-transform_pliki/4b8b31d851e269a8a0a415d02a5b9b11.png"></td>
<td><img class="tex" alt=" \frac{ 1-a z^{-1} \cos( \omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }" src="Z-transform_pliki/688872917fd51180e66767ef557d2a8d.png"></td>
<td><img class="tex" alt=" |z| &gt; |a|\," src="Z-transform_pliki/2a693d4762bb2c8e1e4e11d6aa5ddce7.png"></td>
</tr>
<tr>
<td>21</td>
<td><img class="tex" alt="a^n \sin(\omega_0 n) u[n] \," src="Z-transform_pliki/4fb89703f52df7f80a40801273ed980e.png"></td>
<td><img class="tex" alt=" \frac{ az^{-1} \sin(\omega_0) }{ 1-2az^{-1}\cos(\omega_0)+ a^2 z^{-2} }" src="Z-transform_pliki/95f038d564e100d9cb7eefb8f1a44218.png"></td>
<td><img class="tex" alt=" |z| &gt; |a|\," src="Z-transform_pliki/2a693d4762bb2c8e1e4e11d6aa5ddce7.png"></td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=14" title="Edit section: Relationship to Laplace transform">edit</a>]</span> <span class="mw-headline" id="Relationship_to_Laplace_transform">Relationship to Laplace transform</span></h2>
<p>The <a href="http://en.wikipedia.org/wiki/Bilinear_transform" title="Bilinear transform">Bilinear transform</a>
 is a useful approximation for converting continuous time filters 
(represented in Laplace space) into discrete time filters (represented 
in z space), and vice versa. To do this, you can use the following 
substitutions in <i>H(s)</i> or <i>H(z):</i></p>
<dl>
<dd><img class="tex" alt="\, s =\frac{2}{T} \frac{(z-1)}{(z+1)} \quad " src="Z-transform_pliki/359f1b54214e387ac2d83e6241171aa2.png"></dd>
</dl>
<p>from Laplace to z (Tustin transformation), or</p>
<dl>
<dd><img class="tex" alt="\,  z =\frac{2+sT}{2-sT} \quad  " src="Z-transform_pliki/2f43f96a69b729826defa37ec151fd3e.png"></dd>
</dl>
<p>from z to Laplace. Through the bilinear transformation, the complex 
s-plane (of the Laplace transform) is mapped to the complex z-plane (of 
the z-transform). While this mapping is (necessarily) nonlinear, it is 
useful in that it maps the entire <img class="tex" alt=" j \Omega " src="Z-transform_pliki/b721db425ddbedbab7636231bd9b98ff.png"> axis of the s-plane onto the <a href="http://en.wikipedia.org/wiki/Unit_circle" title="Unit circle">unit circle</a> in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on the <img class="tex" alt=" j \Omega " src="Z-transform_pliki/b721db425ddbedbab7636231bd9b98ff.png"> axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform exists; i.e., that the <img class="tex" alt=" j \Omega " src="Z-transform_pliki/b721db425ddbedbab7636231bd9b98ff.png"> axis is in the region of convergence of the Laplace transform.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=15" title="Edit section: Relationship to Fourier transform">edit</a>]</span> <span class="mw-headline" id="Relationship_to_Fourier_transform">Relationship to Fourier transform</span></h2>
<p>The Z-transform is a generalization of the <a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> (DTFT). The DTFT can be found by evaluating the Z-transform <img class="tex" alt="X(z)\ " src="Z-transform_pliki/985f5e20a8ff19d9596237e1c32f9561.png"> at <img class="tex" alt="z=e^{j\omega}\ " src="Z-transform_pliki/2e9d4c07860a3246b501d39f54aeab9e.png"> (where <img class="tex" alt="\omega" src="Z-transform_pliki/4d1b7b74aba3cfabd624e898d86b4602.png"> is the <a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform#Relationship_to_sampling" title="Discrete-time Fourier transform">normalized frequency</a>) or, in other words, evaluated on the unit circle. In order to determine the <a href="http://en.wikipedia.org/wiki/Frequency_response" title="Frequency response">frequency response</a>
 of the system the Z-transform must be evaluated on the unit circle, 
meaning that the system's region of convergence must contain the unit 
circle. Otherwise, the DTFT of the system does not exist.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=16" title="Edit section: Linear constant-coefficient difference equation">edit</a>]</span> <span class="mw-headline" id="Linear_constant-coefficient_difference_equation">Linear constant-coefficient difference equation</span></h2>
<p>The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the <a href="http://en.wikipedia.org/wiki/Autoregressive_moving_average_model" title="Autoregressive moving average model" class="mw-redirect">autoregressive moving-average</a> equation.</p>
<dl>
<dd><img class="tex" alt="\sum_{p=0}^{N}y[n-p]\alpha_{p} = \sum_{q=0}^{M}x[n-q]\beta_{q}\ " src="Z-transform_pliki/dc56f77192c3d0d2dafc3bf76362f500.png"></dd>
</dl>
<p>Both sides of the above equation can be divided by <img class="tex" alt="\alpha_0 \ " src="Z-transform_pliki/64b09ef92c00dd95c988dc7a55709df4.png">, if it is not zero, normalizing <img class="tex" alt="\alpha_0 = 1\ " src="Z-transform_pliki/cae888a81befbcf2faa9987bfc98f198.png"> and the LCCD equation can be written</p>
<dl>
<dd><img class="tex" alt="y[n] = \sum_{q=0}^{M}x[n-q]\beta_{q} - \sum_{p=1}^{N}y[n-p]\alpha_{p}.\ " src="Z-transform_pliki/26af2334285b94abd5cd6f04bd6cb082.png"></dd>
</dl>
<p>This form of the LCCD equation is favorable to make it more explicit that the "current" output <img class="tex" alt="y[{n}]\ " src="Z-transform_pliki/70d386fe8fc517002af609527dc843e1.png"> is a function of past outputs <img class="tex" alt="y[{n-p}]\ " src="Z-transform_pliki/b7b5966e8e057221625214b957825fbc.png">, current input <img class="tex" alt="x[{n}]\ " src="Z-transform_pliki/f669ef1c4bd8099c03a6b172887ce5eb.png">, and previous inputs <img class="tex" alt="x[{n-q}]\ " src="Z-transform_pliki/2cf5f29f8bd6fcecfc773b18536e913c.png">.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=17" title="Edit section: Transfer function">edit</a>]</span> <span class="mw-headline" id="Transfer_function">Transfer function</span></h3>
<p>Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields</p>
<dl>
<dd><img class="tex" alt="Y(z) \sum_{p=0}^{N}z^{-p}\alpha_{p} = X(z) \sum_{q=0}^{M}z^{-q}\beta_{q}\ " src="Z-transform_pliki/57539ac088ad2948a0ef2934307a3bcf.png"></dd>
</dl>
<p>and rearranging results in</p>
<dl>
<dd><img class="tex" alt="H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{q=0}^{M}z^{-q}\beta_{q}}{\sum_{p=0}^{N}z^{-p}\alpha_{p}} = \frac{\beta_0 + z^{-1} \beta_1 + z^{-2} \beta_2 + \cdots + z^{-M} \beta_M}{\alpha_0 + z^{-1} \alpha_1 + z^{-2} \alpha_2 + \cdots + z^{-N} \alpha_N}.\ " src="Z-transform_pliki/9aa1101fd1b8b2998a84971e9b2e1b4f.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=18" title="Edit section: Zeros and poles">edit</a>]</span> <span class="mw-headline" id="Zeros_and_poles">Zeros and poles</span></h3>
<p>From the <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> the <a href="http://en.wikipedia.org/wiki/Numerator" title="Numerator" class="mw-redirect">numerator</a> has M <a href="http://en.wikipedia.org/wiki/Root_of_a_function" title="Root of a function" class="mw-redirect">roots</a> (corresponding to <a href="http://en.wikipedia.org/wiki/Zero_%28complex_analysis%29" title="Zero (complex analysis)">zeros</a> of H) and the <a href="http://en.wikipedia.org/wiki/Denominator" title="Denominator" class="mw-redirect">denominator</a> has N roots (corresponding to <a href="http://en.wikipedia.org/wiki/Pole_%28complex_analysis%29" title="Pole (complex analysis)">poles</a>). Rewriting the <a href="http://en.wikipedia.org/wiki/Transfer_function" title="Transfer function">transfer function</a> in terms of poles and zeros</p>
<dl>
<dd><img class="tex" alt="H(z) = \frac{(1 - q_1 z^{-1})(1 - q_2 z^{-1})\cdots(1 - q_M z^{-1}) } { (1 - p_1 z^{-1})(1 - p_2 z^{-1})\cdots(1 - p_N z^{-1})}\ " src="Z-transform_pliki/e897a2d9c7748079c93bd5afb40fc7bf.png"></dd>
</dl>
<p>where <img class="tex" alt="q_k\ " src="Z-transform_pliki/e4dae470401c8d3f1d50b756d5fab017.png"> is the <img class="tex" alt="k^{th}\ " src="Z-transform_pliki/ccf97623d498d4ac0c76ab4acacd950f.png"> zero and <img class="tex" alt="p_k\ " src="Z-transform_pliki/360f933ba755fd51a8e6a9ad5470ce48.png"> is the <img class="tex" alt="k^{th}\ " src="Z-transform_pliki/ccf97623d498d4ac0c76ab4acacd950f.png"> pole. The zeros and poles are commonly complex and when plotted on the complex plane (z-plane) it is called the <a href="http://en.wikipedia.org/wiki/Pole-zero_plot" title="Pole-zero plot" class="mw-redirect">pole-zero plot</a>.</p>
<p>In addition, there may also exist zeros and poles at <img class="tex" alt="z=0" src="Z-transform_pliki/8fcd01a17ad602c542f98b916cba57f4.png"> and <img class="tex" alt="z=\infty" src="Z-transform_pliki/6fb3d985fc0f3e3d2ea869467a29df33.png">.
 If we take these poles and zeros as well as multiple-order zeros and 
poles into consideration, the number of zeros and poles are always 
equal.</p>
<p>By factoring the denominator, <a href="http://en.wikipedia.org/wiki/Partial_fraction" title="Partial fraction">partial fraction</a> decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> and the linear constant coefficient difference equation of the system.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=19" title="Edit section: Output response">edit</a>]</span> <span class="mw-headline" id="Output_response">Output response</span></h3>
<p>If such a system <img class="tex" alt="H(z)\ " src="Z-transform_pliki/6a11a82d75b7edaa4a741fae35bff811.png"> is driven by a signal <img class="tex" alt="X(z)\ " src="Z-transform_pliki/985f5e20a8ff19d9596237e1c32f9561.png"> then the output is <img class="tex" alt="Y(z) = H(z)X(z)\ " src="Z-transform_pliki/dfda1920b7cd82eb6120622ca630e6db.png">. By performing <a href="http://en.wikipedia.org/wiki/Partial_fraction" title="Partial fraction">partial fraction</a> decomposition on <img class="tex" alt="Y(z)\ " src="Z-transform_pliki/d4bac27cccb42e25c874a8812b3f92da.png"> and then taking the inverse Z-transform the output <img class="tex" alt="y[n]\ " src="Z-transform_pliki/2bcb339fea06104a3d20c3ccd0c5ef6f.png"> can be found. In practice, it is often useful to fractionally decompose <img class="tex" alt="\frac{Y(z)}{z}\ " src="Z-transform_pliki/d0f5020ec3efbda67e7af64ae106a2a0.png"> before multiplying that quantity by <img class="tex" alt="z\ " src="Z-transform_pliki/02f20ced3d6cc526b4cbc04e5911a5c7.png"> to generate a form of <img class="tex" alt="Y(z)\ " src="Z-transform_pliki/d4bac27cccb42e25c874a8812b3f92da.png"> which has terms with easily computable inverse Z-transforms.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=20" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Advanced_Z-transform" title="Advanced Z-transform">Advanced Z-transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Star_transform" title="Star transform">Star transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Bilinear_transform" title="Bilinear transform">Bilinear transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Finite_impulse_response" title="Finite impulse response">Finite impulse response</a></li>
<li><a href="http://en.wikipedia.org/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li>
<li><a href="http://en.wikipedia.org/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Laurent_series" title="Laurent series">Laurent series</a></li>
<li><a href="http://en.wikipedia.org/wiki/Probability-generating_function" title="Probability-generating function">Probability-generating function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Zeta_function_regularization" title="Zeta function regularization">Zeta function regularization</a></li>
<li><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Difference_equation" title="Difference equation" class="mw-redirect">Difference equation</a> (recurrence relation)</li>
<li><a href="http://en.wikipedia.org/wiki/Convolution#Discrete_convolution" title="Convolution">Discrete convolution</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=21" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<div class="reflist" style="list-style-type: decimal;">
<ol class="references">
<li id="cite_note-0"><b><a href="#cite_ref-0">^</a></b> <span class="reference-text"><span class="citation book">E. R. Kanasewich (1981). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=k8SSLy-FYagC&amp;pg=PA185"><i>Time sequence analysis in geophysics</i></a> (3rd ed.). University of Alberta. pp.&nbsp;185–186. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-88864-074-1" title="Special:BookSources/978-0-88864-074-1">978-0-88864-074-1</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://books.google.com/books?id=k8SSLy-FYagC&amp;pg=PA185">http://books.google.com/books?id=k8SSLy-FYagC&amp;pg=PA185</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Time+sequence+analysis+in+geophysics&amp;rft.aulast=E.+R.+Kanasewich&amp;rft.au=E.+R.+Kanasewich&amp;rft.date=1981&amp;rft.pages=pp.%26nbsp%3B185%E2%80%93186&amp;rft.edition=3rd&amp;rft.pub=University+of+Alberta&amp;rft.isbn=978-0-88864-074-1&amp;rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dk8SSLy-FYagC%26pg%3DPA185&amp;rfr_id=info:sid/en.wikipedia.org:Z-transform"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text"><span class="citation Journal">J. R. Ragazzini and L. A. Zadeh (1952). "The analysis of sampled-data systems". <i>Trans. Am. Inst. Elec. Eng.</i> <b>71</b> (II): 225–234.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=The+analysis+of+sampled-data+systems&amp;rft.jtitle=Trans.+Am.+Inst.+Elec.+Eng.&amp;rft.aulast=J.+R.+Ragazzini+and+L.+A.+Zadeh&amp;rft.au=J.+R.+Ragazzini+and+L.+A.+Zadeh&amp;rft.date=1952&amp;rft.volume=71&amp;rft.issue=II&amp;rft.pages=225%E2%80%93234&amp;rfr_id=info:sid/en.wikipedia.org:Z-transform"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"><span class="citation book">Cornelius T. Leondes (1996). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=aQbk3uidEJoC&amp;pg=PA123"><i>Digital control systems implementation and computational techniques</i></a>. Academic Press. p.&nbsp;123. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-12-012779-5" title="Special:BookSources/978-0-12-012779-5">978-0-12-012779-5</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://books.google.com/books?id=aQbk3uidEJoC&amp;pg=PA123">http://books.google.com/books?id=aQbk3uidEJoC&amp;pg=PA123</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Digital+control+systems+implementation+and+computational+techniques&amp;rft.aulast=Cornelius+T.+Leondes&amp;rft.au=Cornelius+T.+Leondes&amp;rft.date=1996&amp;rft.pages=p.%26nbsp%3B123&amp;rft.pub=Academic+Press&amp;rft.isbn=978-0-12-012779-5&amp;rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaQbk3uidEJoC%26pg%3DPA123&amp;rfr_id=info:sid/en.wikipedia.org:Z-transform"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><span class="citation book">Eliahu Ibrahim Jury (1958). <i>Sampled-Data Control Systems</i>. John Wiley &amp; Sons.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Sampled-Data+Control+Systems&amp;rft.aulast=Eliahu+Ibrahim+Jury&amp;rft.au=Eliahu+Ibrahim+Jury&amp;rft.date=1958&amp;rft.pub=John+Wiley+%26+Sons&amp;rfr_id=info:sid/en.wikipedia.org:Z-transform"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="reference-text"><span class="citation book">Eliahu Ibrahim Jury (1973). <i>Theory and Application of the Z-Transform Method</i>. Krieger Pub Co. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-88275-122-0" title="Special:BookSources/0-88275-122-0">0-88275-122-0</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Theory+and+Application+of+the+Z-Transform+Method&amp;rft.aulast=Eliahu+Ibrahim+Jury&amp;rft.au=Eliahu+Ibrahim+Jury&amp;rft.date=1973&amp;rft.pub=Krieger+Pub+Co&amp;rft.isbn=0-88275-122-0&amp;rfr_id=info:sid/en.wikipedia.org:Z-transform"><span style="display: none;">&nbsp;</span></span></span></li>
<li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text"><span class="citation book">Eliahu Ibrahim Jury (1964). <i>Theory and Application of the Z-Transform Method</i>. John Wiley &amp; Sons. p.&nbsp;1.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Theory+and+Application+of+the+Z-Transform+Method&amp;rft.aulast=Eliahu+Ibrahim+Jury&amp;rft.au=Eliahu+Ibrahim+Jury&amp;rft.date=1964&amp;rft.pages=p.%26nbsp%3B1&amp;rft.pub=John+Wiley+%26+Sons&amp;rfr_id=info:sid/en.wikipedia.org:Z-transform"><span style="display: none;">&nbsp;</span></span></span></li>
</ol>
</div>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=22" title="Edit section: Further reading">edit</a>]</span> <span class="mw-headline" id="Further_reading">Further reading</span></h2>
<ul>
<li>Refaat El Attar, <i>Lecture notes on Z-Transform</i>, Lulu Press, Morrisville NC, 2005. <a href="http://en.wikipedia.org/wiki/Special:BookSources/141161979X" class="internal mw-magiclink-isbn">ISBN 1-4116-1979-X</a>.</li>
<li>Ogata, Katsuhiko, <i>Discrete Time Control Systems 2nd Ed</i>, Prentice-Hall Inc, 1995, 1987. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0130342815" class="internal mw-magiclink-isbn">ISBN 0-13-034281-5</a>.</li>
<li>Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal
 Processing, 2nd Edition, Prentice Hall Signal Processing Series. <a href="http://en.wikipedia.org/wiki/Special:BookSources/0137549202" class="internal mw-magiclink-isbn">ISBN 0-13-754920-2</a>.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Z-transform&amp;action=edit&amp;section=23" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a rel="nofollow" class="external text" href="http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/LaplaceZTable/LaplaceZFuncTable.html">Z-Transform table of some common Laplace transforms</a></li>
<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Z-Transform.html">Mathworld's entry on the Z-transform</a></li>
<li><a rel="nofollow" class="external text" href="http://www.dsprelated.com/comp.dsp/keyword/Z_Transform.php">Z-Transform threads in Comp.DSP</a></li>
<li><a rel="nofollow" class="external text" href="http://math.fullerton.edu/mathews/c2003/ZTransformIntroMod.html">Z-Transform Module by John H. Mathews</a></li>
</ul>
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<li><a href="http://en.wikipedia.org/wiki/Estimation_theory" title="Estimation theory">estimation theory</a></li>
<li><a href="http://en.wikipedia.org/wiki/Detection_theory" title="Detection theory">detection theory</a></li>
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<li><a href="http://en.wikipedia.org/wiki/Audio_signal_processing" title="Audio signal processing">audio signal processing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Digital_image_processing" title="Digital image processing">digital image processing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Speech_processing" title="Speech processing">speech processing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Statistical_signal_processing" title="Statistical signal processing">statistical signal processing</a></li>
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<th scope="row" class="navbox-group" style="">Techniques</th>
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<li><a href="http://en.wikipedia.org/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)</li>
<li><a href="http://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)</li>
<li><a href="http://en.wikipedia.org/wiki/Impulse_invariance" title="Impulse invariance">Impulse invariance</a></li>
<li><a href="http://en.wikipedia.org/wiki/Bilinear_transform" title="Bilinear transform">bilinear transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Pole%E2%80%93zero_mapping" title="Pole–zero mapping" class="mw-redirect">pole–zero mapping</a></li>
<li><strong class="selflink">Z-transform</strong></li>
<li><a href="http://en.wikipedia.org/wiki/Advanced_Z-transform" title="Advanced Z-transform">advanced Z-transform</a></li>
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<tr>
<th scope="row" class="navbox-group" style=""><a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling</a></th>
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<li><a href="http://en.wikipedia.org/wiki/Oversampling" title="Oversampling">oversampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Undersampling" title="Undersampling">undersampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Downsampling" title="Downsampling">downsampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Upsampling" title="Upsampling">upsampling</a></li>
<li><a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">aliasing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing filter</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sampling_rate" title="Sampling rate">sampling rate</a></li>
<li><a href="http://en.wikipedia.org/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a>/<a href="http://en.wikipedia.org/wiki/Nyquist_frequency" title="Nyquist frequency">frequency</a></li>
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			<li class="interwiki-ca"><a href="http://ca.wikipedia.org/wiki/Transformada_Z" title="Transformada Z" hreflang="ca" lang="ca">Català</a></li>
			<li class="interwiki-cs"><a href="http://cs.wikipedia.org/wiki/Z-transformace" title="Z-transformace" hreflang="cs" lang="cs">Česky</a></li>
			<li class="interwiki-de"><a href="http://de.wikipedia.org/wiki/Z-Transformation" title="Z-Transformation" hreflang="de" lang="de">Deutsch</a></li>
			<li class="interwiki-es"><a href="http://es.wikipedia.org/wiki/Transformada_Z" title="Transformada Z" hreflang="es" lang="es">Español</a></li>
			<li class="interwiki-fa"><a href="http://fa.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%DB%8C%D9%84_%D8%B2%D8%AF" title="تبدیل زد" hreflang="fa" lang="fa">فارسی</a></li>
			<li class="interwiki-fr"><a href="http://fr.wikipedia.org/wiki/Transform%C3%A9e_en_Z" title="Transformée en Z" hreflang="fr" lang="fr">Français</a></li>
			<li class="interwiki-it"><a href="http://it.wikipedia.org/wiki/Trasformata_zeta" title="Trasformata zeta" hreflang="it" lang="it">Italiano</a></li>
			<li class="interwiki-he"><a href="http://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%9E%D7%A8%D7%AA_Z" title="התמרת Z" hreflang="he" lang="he">עברית</a></li>
			<li class="interwiki-nl"><a href="http://nl.wikipedia.org/wiki/Z-transformatie" title="Z-transformatie" hreflang="nl" lang="nl">Nederlands</a></li>
			<li class="interwiki-ja"><a href="http://ja.wikipedia.org/wiki/Z%E5%A4%89%E6%8F%9B" title="Z変換" hreflang="ja" lang="ja">日本語</a></li>
			<li class="interwiki-pl"><a href="http://pl.wikipedia.org/wiki/Transformata_Z" title="Transformata Z" hreflang="pl" lang="pl">Polski</a></li>
			<li class="interwiki-pt"><a href="http://pt.wikipedia.org/wiki/Transformada_Z" title="Transformada Z" hreflang="pt" lang="pt">Português</a></li>
			<li class="interwiki-ro"><a href="http://ro.wikipedia.org/wiki/Transformata_Z" title="Transformata Z" hreflang="ro" lang="ro">Română</a></li>
			<li class="interwiki-ru"><a href="http://ru.wikipedia.org/wiki/Z-%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5" title="Z-преобразование" hreflang="ru" lang="ru">Русский</a></li>
			<li class="interwiki-sd"><a href="http://sd.wikipedia.org/wiki/%D8%B2%D9%8A%DA%8A_%D9%85%D8%A8%D8%AF%D9%84" title="زيڊ مبدل" hreflang="sd" lang="sd">سنڌي</a></li>
			<li class="interwiki-fi"><a href="http://fi.wikipedia.org/wiki/Z-muunnos" title="Z-muunnos" hreflang="fi" lang="fi">Suomi</a></li>
			<li class="interwiki-sr"><a href="http://sr.wikipedia.org/wiki/Z-%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Z-трансформација" hreflang="sr" lang="sr">Српски / Srpski</a></li>
			<li class="interwiki-sv"><a href="http://sv.wikipedia.org/wiki/Z-transform" title="Z-transform" hreflang="sv" lang="sv">Svenska</a></li>
			<li class="interwiki-tr"><a href="http://tr.wikipedia.org/wiki/Z-d%C3%B6n%C3%BC%C5%9F%C3%BCm%C3%BC" title="Z-dönüşümü" hreflang="tr" lang="tr">Türkçe</a></li>
			<li class="interwiki-zh"><a href="http://zh.wikipedia.org/wiki/Z%E8%BD%89%E6%8F%9B" title="Z轉換" hreflang="zh" lang="zh">中文</a></li>
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